Introduction to Automata
The methods and the madness
What is the study of Automata
Theory?
• The study of abstract computing devices, or “machines.”
• Days before digital computers
– What is possible to compute with an abstract machine
– Seminal work by Alan Turing
• Why is this useful?
– Direct application to creating compilers, programming languages,
designing applications.
– Formal framework to analyze new types of computing devices, e.g.
biocomputers or quantum computers.
– Develop mathematically mature computer scientists capable of
precise and formal reasoning!
• 5 major topics in Automata Theory
Finite State Automata
• Deterministic and non-deterministic finite
state machines
• Regular expressions and languages.
• Techniques for identifying and describing
regular languages; techniques for showing
that a language is not regular. Properties of
such languages.
Context-Free Languages
• Context-free grammars, parse trees
• Derivations and ambiguity
• Relation to pushdown automata. Properties
of such languages and techniques for
showing that a language is not context-free.
Turing Machines
• Basic definitions and relation to the notion
of an algorithm or program.
• Power of Turing Machines.
Undecidability and Complexity
• Undecidability
– Recursive and recursively enumerable languages.
– Universal Turing Machines.
– Limitations on our ability to compute; undecidable problems.
• Computational Complexity
–
–
–
–
Decidable problems for which no sufficient algorithms are known.
Polynomial time computability.
The notion of NP-completeness and problem reductions.
Examples of hard problems.
• Let’s start with a big-picture overview of these 5 topics
Finite State Automata
• Automata – plural of “automaton”
– i.e. a robot
• Finite state automata then a “robot composed of a finite
number of states”
– Informally, a finite list of states with transitions between the states
• Useful to model hardware, software, algorithms, processes
–
–
–
–
Software to design and verify circuit behavior
Lexical analyzer of a typical compiler
Parser for natural language processing
An efficient scanner for patterns in large bodies of text (e.g. text
search on the web)
– Verification of protocols (e.g. communications, security).
On-Off Switch Automaton
• Here is perhaps one of the
simplest finite automaton, an onoff switch
• States are represented by circles.
Generally we will use much
more generic names for states
(e.g. q1, q2). Edges or arcs
between states indicate
transitions or inputs to the
system. The “start” edge
indicates which state we start in.
• Sometimes it is necessary to
indicate a “final” or “accepting”
state. We’ll do this by drawing
the state in double circles
P u sh
S ta rt
On
O ff
P u sh
Automata Example
• Consider an automaton to parse an HTML
document that attempts to identify title-author
pairs in a bulleted or ordered list. This might be
useful to generate a reading list of some sort
automatically.
• Example:
– <ul><li>Computation by Michael Sipser</li></ul>
• A hypothetical automaton to address this task is
shown next that scans for the letters “by” inside a
list item
Author Scanning Automaton
Gas Furnace Example
F urnace
R
W
To
T herm o stat
G
• The R terminal is the hot wire and completes a
circuit. When R and G are connected, the blower
turns on. When R and W are connected, the burner
comes on. Any other state where R is not connected
to either G or W results in no action.
Furnace Automaton
• Could be implemented in a thermostat
R+W
S tart
B l O ff
B r O ff
B l O ff
Br On
R -W
R -G
R+G
R -G
R+G
R+W
Bl On
B r O ff
Bl On
Br On
R -W
Furnace Notes
• We left out connections that have no effect
– E.g. connecting W and G
• Once the logic in the automata has been formalized, the
model can be used to construct an actual circuit to control
the furnace (i.e., a thermostat).
• The model can also help to identify states that may be
dangerous or problematic.
– E.g. state with Burner On and Blower Off could overhead the
furnace
– Want to avoid this state or add some additional states to prevent
failure from occurring (e.g., a timeout or failsafe )
Languages and Grammars
• Languages and grammars provide a different
“view” of computing than automata
– Often languages and grammars are identical to
automata! This will be a central theme we will revisit
several times.
• Consider the HTML checking automata
– Instead of a set of states, we can view this as the
problem of determining all of the strings that make up
valid author/title pairs.
– The set of all valid strings accepted by the automata
makes up the Language for this particular problem
Grammar Example
• Just like English, languages can be described by grammars. For example,
below is a very simple grammar:
S Noun Verb-Phrase
Verb-Phrase  Verb Noun
Noun  { Kenrick, cows }
Verb  { loves, eats }
• Using this simple grammar our language allows the following sentences.
They are “in” the Language defined by the grammar:
Kenrick loves Kenrick
Kenrick loves cows
Kenrick eats Kenrick
• Some sentences not in the grammar:
Kenrick eats cows
Kenrick loves cows and kenrick.
Cows loves Kenrick
Cows eats love cows.
Cows loves cows
Kenrick loves chocolate.
Cows eats Kenrick
Cows eats cows
Grammars and Languages
• Later we’ll see ways to go back and forth between a
grammar-based definition for languages and an automata
based definition
• Like a game, given a sentence (we’ll call this a “string”)
determine if it is in or out of the Language
• Grammar provides a “cut” through the space of possible
strings – will go from crude to sophisticated cuts
K enrick loves K enrick cow s
cow s eat eat loves
O ut
In
K enrick loves K enrick
K enrick loves cow s
Mathematical Notions
• Skipping these topics, but they’re briefly
described in the textbook
– Sets
• Empty set, subset, union, Venn diagram, etc.
– Sequences
– Tuples
– Functions and Relations
• Mapping from Domain to Range
– Boolean Logic
Graphs
• A graph is composed of edges E and vertices V
that link the nodes together. A graph G is often
denoted G=(V,E) where V is the set of vertices and
E the set of edges.
• Two types of graphs:
– Directed graphs: G=(V,E) where E is composed of
ordered pairs of vertices; i.e. the edges have direction
and point from one vertex to another.
– Undirected graphs: G=(V,E) where E is composed of
unordered pairs of vertices; i.e. the edges are
bidirectional.
Directed Graph
1
2
5
3
4
Undirected Graph
1
2
5
3
4
Graph Terminology
• The degree of a vertex in an undirected graph is
the number of edges that leave/enter the vertex.
• The degree of a vertex in a directed graph is the
same, but we distinguish between in-degree and
out-degree. Degree = in-degree + out-degree.
• A path from u to v is <u, w1, …v> and
(u,w1)(w1,w2)(w2,w3)…(wn,v)
• The running time of a graph algorithm expressed
in terms of E and V, where E = |E| and V=|V|; e.g.
G=O(EV) is |E| * |V|
Graph Terminology
• A path in a graph is a sequence of nodes connected by edges
• A graph is connected if every two nodes have a path
between them
• A path is a cycle if it starts and ends in the same node
– A simple cycle is one that contains at least three nodes and repeats
only the first and last nodes
• A graph is a tree if it is connected and has no simple cycles
• A graph is strongly connected if a path connects every two
nodes
Language Definitions (1)
•
An alphabet is a finite, nonempty set of
symbols. By convention we use the
symbol  for an alphabet.
–
–
In the previous example, our alphabet
consisted of words, but normally our alphabet
will consist of individual characters.
Examples
•
•
 = {0,1} the binary alphabet
 = {a,b, … z} the set of all lowercase letters
Language Definitions (2)
• string (or sometimes a word)
– A finite sequence of symbols chosen from an alphabet.
For example, 010101010 is a string chosen from the
binary alphabet, as is the string 0000 or 1111.
• The empty string is the string with zero
occurrences of symbols. This string is denoted ε
and may be chosen from any alphabet.
• The power notation is used to represent multiple
occurrences of a string; e.g. a3 = aaa, a2 = aa, etc.
Language Definitions (3)
• The length of a string indicates how many
symbols are in that string.
– E.g., the string 0101 using the binary alphabet
has a length of 4.
– The standard notation for a string w is to use
|w|. For example, |0101| is 4.
Language Definitions (4)
• Powers of an alphabet
– If  is an alphabet, we can express the set of all strings of a certain length
from that alphabet by using an exponential notation.
– k is defined to be the set of strings of length k, each of whose symbols is
in .
• For example, given the alphabet  = {0,1,2} then:
–
–
–
–
0 = {ε}
1 = {0,1,2}
2 = {00,01,02,10,11,12,20,21,22}
3 = {000,001,002,... 222}
• Note that  and 1 are different. The first is the alphabet; its
members are 0,1,2. The second is the set of strings whose
members are the strings 0,1,2, each a string of length 1.
• By convention, we will try to use lower-case letters at the
beginning of the alphabet to denote symbols, and lower-case
letters near the end of the alphabet to represent strings.
Language Definitions (5)
• Set of all Strings
– The set of all strings over an alphabet is denoted by
*. That is:

*
       ...
0
1
2
– Sometimes it is useful to exclude the empty string from
the set of strings. The set of nonempty strings from the
alphabet is denoted by +.
Language Definitions (6)
• To concatenate strings, we will simply put
them right next to one another.
• Example:
– If x and y are strings, where x=001 and y=111
then xy = 001111
– For any string w, the equation εw = wε = w.
Formal Definition of Languages
• We have finally covered enough definitions
to formally define a language!
• A Language
– A set of strings all of which are chosen from
some * is called a language.
– If  is an alphabet and L is a subset of * then
L is a language over .
– Note that a language need not include all strings
in *.
Language Examples
1. The language of all strings consisting of n 0’s
followed by n 1’s, for some n0: { ε, 01, 0011,
000111, …}
2. The set of binary numbers whose value is a
prime: { 10, 11, 101, 111, …}
3. Ø is the empty language, which is a language
over any alphabet.
4. {ε} is the language consisting of only the empty
string. Note that this is not the same as example
#3, the former has no strings and the latter has
one string.
Language Definition - Problem
• A problem is the question of deciding whether a
given string is a member of some particular
language.
– More colloquially, a problem is expressed as
membership in the language.
– Languages and problems are basically the same thing.
When we care about the strings, we tend to think of it
as a language. When we assign semantics to the
strings, e.g. maybe the strings encode graphs, logical
expressions, or integers, then we will tend to think of
the set of strings as a solution to the problem.
Set-Forming Notation
• A notation we will commonly use to define languages is by
a “set-former”:
{ w | something about w }
• The expression is read “the set of words w such that
(whatever is said about w to the right of the vertical bar).”
• For example:
–
–
–
–
{w | w consists of an equal number of 0’s and 1’s }.
{w | w is a binary integer that is prime }
{ 0n1n | n >=1 }. This includes 01, 0011, 000111, etc. but not ε
{ 0n1 | n>=0 }. This includes 1, 01, 001, 0001, 00001, etc.
Bigger Picture
• Finite state automata provide only a crude “cut” of * to
select the strings we will accept.
• Turing machines and more complex grammars provide for
more sophisticated ways to define the language. One way
this will be accomplished is there will no longer be a finite
set of states, but an infinite number of possible states.
C rude
e.g. S im ple autom ata
C o m plex
e.g. T uring M achine
Taxonomy of Complexity
U nco m putable
T uring M achines
P hrase S tructure
L inear bounded
autom ata
C ontext-S ensitive
P ushdow n
autom ata
C ontext-F ree
F inite state
autom ata
R egular
M achines
C o m plex
C rude
G ram m ars/L anguages
Complexity and Uncomputability
•
Complexity is the study of the limits of computation.
There are two important issues:
1.
2.
Decidability. What can a computer do at all? The problems that
can be solved by a computer in a realistic amount of time are
called decidable. Some problems are undecidable, or only semidecidable (e.g. membership in certain languages, must
enumerate, but may be infinite)
Intractability. What can a computer do efficiently? This studies
the problems that can be solved by a computer using no more
time than some slowly growing function of the size of the input.
Typically we will take all polynomial functions to be tractable,
while functions that grow faster than polynomial intractable.
Complexity
Hierarchy
Introduction to Formal Proof
• In this class, sometimes we will give formal
proofs and at other times intuitive “proofs”
• Mostly inductive proofs
• First, a bit about deductive proofs
Deductive Proofs
• Given a hypothesis H, and some statements, generate a
conclusion C
• Sherlock Holmes style of reasoning
• Example: consider the following theorem
– If x 4 then 2x  x2
– Here, H is x  4 and C is 2x  x2
– Intuitive deductive proof
• Each time x increases by one, the left hand side doubles in size.
However, the right side increases by the ratio ((x+1)/x)2. When x=4,
this ratio is 1.56. As x increases and approaches infinity, the ratio
((x+1)/x)2 approaches 1. This means the ratio gets smaller as x
increases. Consequently, 1.56 is the largest that the right hand side will
increase. Since 1.56 < 2, the left side is increasing faster than the right
side
Basic Formal Logic (1)
• An “If H then C” statement is typically expressed
as:
HC
or
H implies C
• The logic truth table for implication is:
H
C
H C (i.e. ¬H  C)
F
F
T
F
T
T
T
F
F
T
T
T
Basic Formal Logic (2)
• If and Only If statements, e.g. “If and only if H then C”
means that HC and C H.
• Sometimes this will be written as
HC or “H iff C”.
The truth table is:
H
C
HC (i.e. H equals C)
F
F
T
F
T
F
T
F
F
T
T
T
Modus Ponens
• modus ponens (Latin for “method of affirming”') can be
used to form chains of logic to reach a desired conclusion.
• In other words, given:
HC
and
H
Then we can infer C
• Example: given “If Joe and Sally are siblings then Joe and
Sally are related” as a true assertion, and also given “Joe
and Sally are siblings” as a true assertion, then we can
conclude “Joe and Sally are related.”
Modus Tollens
• modus tollens (Latin for “method of denying”'). This reasons
backwards across the implication.
– Cognitive psychologists have shown that under 60% of college students
have a solid intuitive understanding of Modus Tollens versus almost
100% for Modus Ponens
• If we are given:
HC
and
¬C
then we can infer ¬H.
• For example, given: “If Joe and Sally are siblings then Joe and Sally
are related” as a true assertion, and also given “Joe and Sally are not
related” as a true assertion, then we can conclude “Joe and Sally are
not siblings.”
– What if we are told Joe and Sally are not siblings? Can we conclude
anything?
Short Exercises (1)
• If Elvis is the king of rock and roll, then
Elvis lives. Elvis is the king of rock and
roll. Therefore Elvis is alive. Valid or
invalid?
– This argument is valid, in that the conclusion is
established (by Modus ponens) if the premises
are true. However, if you consider the first
premise to be false (unless you live in Vegas)
then the conclusion is false.
Short Exercises (2)
• If the stock market keeps going up, then I'm
going to get rich. The stock market isn't
going to keep going up. Therefore I'm not
going to get rich. Valid or invalid?
– This argument is invalid, specifically an inverse
error. Its form is from ¬H and infer ¬C. This
yields an inverse error.
Short Exercises (3)
• If New York is a big city, then New York
has lots of people. New York has lots of
people. Therefore New York is a big city.
Valid or invalid?
– This argument is invalid, even though the
conclusion is true. We are given HC and
given C. This does not mean that CH so we
can’t infer H is true.
Proof by Contradiction
•
•
Suppose that we want to prove H and we know that C is true. Instead of
proving H directly, we may instead show that assuming ¬H leads to a
contradiction. Therefore H must be true.
Example:
–
A large sum of money has been stolen from the bank. The criminal(s) were
seen driving away from the scene. From questioning criminals A, B, and C we
know:
•
•
•
–
Turned into logical statements
1.
2.
3.
–
No one other than A, B, or C were involved in the robbery.
C never pulls a job without A
B does not know how to drive
AB C
CA
B  (A  C)
A, B, or C is guilty
If C is guilty, A is also guilty
If B is guilty, A or C is guilty
Is A innocent or guilty? Let’s assume that A is innocent, i.e.:
i. ¬A
ii. From ¬A and #2 using modus tollens, we can infer ¬C
iii. We thus have ¬A  ¬C, which by De Morgan’s Law is logically equivalent to ¬(A 
C)
iv. From ¬(A  C) and #3 using modus tollens, we can infer ¬B
v. We now have ¬A and ¬B and ¬C which contradicts assumption #1! A is guilty.
Proof by Contrapositive
• Proof by contrapositive takes advantage of the
logical equivalence between "H implies C" and
"Not C implies Not H".
• For example, the assertion "If it is my car, then it
is red" is equivalent to "If that car is not red, then
it is not mine".
• To prove "If P, Then Q" by the method of
contrapositive means to prove "If Not Q, Then Not
P".
Contrapositive Example
• An integer x is called even (respectively
odd) if there is another integer k for which
x = 2k (respectively 2k+1).
• Two integers are said to have the same
parity if they are both odd or both even.
• Theorem. If x and y are two integers for
which x+y is even, then x and y have the
same parity
Contrapositive Example
• Proof of the theorem
– The contrapositive version of this theorem is "If x and y
are two integers with opposite parity, then their sum
must be odd."
– Assume x and y have opposite parity.
• Since one of these integers is even and the other odd, there is
no loss of generality to suppose x is even and y is odd.
– Thus, there are integers k and m for which x = 2k and y
= 2m+1. Then, we compute the sum x+y = 2k + 2m + 1
= 2(k+m) + 1, which is an odd integer by definition.
Contrapositive vs. Contradiction
• Both methods somewhat similar, but different.
• In contrapositive, we assume ¬C and prove ¬H,
given HC.
– The method of Contrapositive has the advantage that
your goal is clear: Prove Not H.
• In the method of Contradiction, your goal is to
prove a contradiction, but it is not always clear
what the contradiction is going to be at the start.
– Indeed, one may never be found (and will never be
found if the hypothesis is false).
Proof by Induction
•
•
•
Essential for proving recursively defined objects
We can perform induction on integers, automata,
and concepts like trees or graphs.
To make an inductive proof about a statement
S(X) we need to prove two things:
1. Basis: Prove for one or several small values of X
directly.
2. Inductive step: Assume S(Y ) for Y “smaller than" X;
then prove S(X) using that assumption.
Familiar Induction Example?
• For all n  0, prove that:
n
n ( n  1)
i 1
2
i
• First prove the basis. We pick n=0. When n=0,
there is a general principle that when the upper
limit (0) of a sum is less than the lower limit (1)
then the sum is over no terms and therefore the
0
sum is 0. That is:
i0
i 1
Familiar Induction Example
• Next prove the induction. Assume n  0. We must prove
that the theorem implies the same formula when n is larger.
For integers, we will use n+1 as the next largest value.
This means that the formula should hold with n +1
substituted for n: n 1
( n  1)( n  2 )
i
2
i 1
n  3n  2
2

2
• This should equal what we came up with previously if we just add on
an extra n+1 term:
n 1
 n 
 i    i   n  1
i 1
 i 1 
Familiar Induction Example
n 1
 n 
 i    i   n  1
i 1
 i 1 
• Continued:
 n 
  i   n  1
 i 1 

n ( n  1)
 ( n  1)
2
n n
2


2n  2
2
2
n  3n  2
2

2
This matches what we got from
the inductive step, and the
proof is complete.
Second Induction Example
• If x 4 then 2x  x2
• Basis: If x=4, then 2x is 16 and x2 is 16.
Thus, the theorem holds.
• Induction: Suppose for some x 4 that 2x 
x2. With this statement as the hypothesis,
we need to prove the same statement, with
x+1 in place of x: 2(x+1)  (x+1)2
Second Induction Example
• 2(x+1)  (x+1)2
?
• Rewrite in terms of S(x)
– 2(x+1) = 2*2x
– We are assuming 2x  x2
– So therefore 2(x+1) = 2*2x  2x2
• Substitute (ii) into (i)
–
–
–
–
–
(i)
(ii)
2x2  (x+1)2
2x2  (x2+2x+1)
x2  2x+1
x  2 + 1/x
Since x >=4, we get some value >=4 on the left side. The right
side will equal at most 2.25 and in fact gets smaller and
approaches 2 as x increases. Consequently, we have proven the
theorem to be true by induction.